$*$The following is OPTIONAL$-$ Do not turn in $4.$ Talisman Games, a local gaming store, stocks the $4$th edition of the rulebook of a popular pen$-$and$-$paper role$-$playing game Daggers and Dungeons. Talisman orders the rulebooks directly from its publisher$-$ Wizards of the Shore $($WotS$).$ Talisman orders a rulebook at a wholesale price of S$1$S and sells it for $$20.$ The lead$-$time from the supplier is $2$ weeks and there is a fixed order cost of S$10.$ Demand for the rulebook each week is independent of demand in previous week standard deviation of $3$ per week. The owners of the store estimate that their opportunity cost of capital is around $15\%$ per year. s and is approximately normal with a mean of $4$ per week with If a customer comes looking to buy a rule book but finds Talisman is stocked out, their copy is backordered and picked up when the next shipment arrives at Talisman from WotS. The customers are always eager to pick up their book, so no sales are lost. To reward loyalty, Talisman throws in a free set of dice $($worth $$2)$ with every backordered book purchase. $[4]$ What is Talisman's optimal inventory control policy; that is$,$ its optimal order quantity O$*$ and re$-$order point R$.?$ a$.12]$ How long does the average rulebook spend in Talisman before it is sold to a customer? b$.$ This morning a sales rep from WotS contacted the owners of Talisman. The owners learn that the new $5$th edition of Daggers and Dungeons will be released in exactly $12$ weeks. The sales rep told them that today was their last chance to order $4$th edition books before they go out of print. Talisman forecasts that demand for the $4$th edition will stay at the same level as before at its original price up until the $5$th edition is released, but afterwards they will need to discount the $4$th edition book by $30\%$ in order to sell the remaining units $($assume all are eventually sold at this reduced price$).$ c$.[4]$ What inventory level for $4$h edition books should Talisman target to have in stock when the last order arrives two weeks from now?